Heights and quantitative arithmetic on stacky curves

TL;DR

This paper explores heights on stacky curves, constructs a dual height, counts rational points, and connects height properties to conjectures like Vojta and the abc conjecture, revealing new insights in arithmetic geometry.

Contribution

It introduces an elementary dual height construction, counts rational points with bounded height, and links height properties to major conjectures in the context of stacky curves.

Findings

Counted rational points with bounded height on a specific stacky curve.

Showed the failure of Northcott property for certain heights when Euler characteristic is non-positive.

Established equivalence between a Vojta conjecture variant and the abc conjecture for stacky curves.

Abstract

In this paper we investigate a family of algebraic stacks, the so-called stacky curves, in the context of the general theory of heights on algebraic stacks due to Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. Next we count rational points having bounded E-S-ZB height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We then show that when the Euler characteristic of stacky curves is non-positive, that the E-S-ZB height coming from the anti-canonical divisor class fails to have the Northcott property. Next we prove a generalized version of a conjecture of Vojta, applied to stacky curves with negative Euler characteristic and coarse space $\mathbb{P}^1$, is equivalent to the $abc$-conjecture. Finally, we prove that in the negative characteristic case the purely "stacky" part of the E-S-ZB height exhibits the Northcott property.